6. Normal Distribution & Continuous Random Variables

Uniform Distribution

6. Normal Distribution & Continuous Random Variables

Uniform Distribution

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Multiple Choice

Shade the area corresponding to the probability listed, then find the probability.
$P\left(X<7.5\right)$

Graph showing a shaded area under a uniform distribution curve, indicating P(X<7.5) with axes labeled for probability density and x. ; $P\left(X<7.5\right)=0.15$

Graph showing a uniform distribution with shaded area representing P(X<7.5) under the probability density curve. ; $P\left(X<7.5\right)=0.25$

Graph showing a uniform distribution with shaded area representing P(X<7.5) under the probability density curve. ; $P\left(X<7.5\right)=0.5625$

Graph showing a shaded area under a uniform distribution curve, indicating P(X<7.5) with axes labeled for probability density and x. ; $P\left(X<7.5\right)=0.5625$

Verified step by step guidance

Step 1: Identify the type of probability distribution represented in the graph. The graph shows a triangular probability density function (PDF), which is a continuous distribution. The area under the curve represents probabilities.

Step 2: Recognize that the problem asks for the probability P(X < 7.5). This corresponds to the shaded area under the curve from the leftmost point (x = 2.5) to x = 7.5.

Step 3: Break the shaded area into geometric shapes for calculation. The shaded region forms a triangle. The base of the triangle spans from x = 2.5 to x = 7.5, and the height corresponds to the value of the PDF at x = 7.5.

Step 4: Use the formula for the area of a triangle: Area = (1/2) × base × height. The base is (7.5 - 2.5) = 5, and the height can be determined from the graph as 0.15 (the value of the PDF at x = 7.5).

Step 5: Calculate the area using the formula. This area represents the probability P(X < 7.5). Substitute the values into the formula: Area = (1/2) × 5 × 0.15. This will give the probability value.

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Shade the area corresponding to the probability listed, then find the probability.P(X<7.5)P\left(X<7.5\right)P(X<7.5)

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Shade the area corresponding to the probability listed, then find the probability.
$P\left(X<7.5\right)$